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What evidence do we have that Buddhism ceases suffering? It's been around for several millennia.
Well, you need to try it for yourself and see. It takes a fair bit of mucking around and advice/assistance from more experienced people to get it right, but I know from experience that it's effective.
one question, may be a stupid question, but still asking - in above 6 words, there is No Intention.. - the question is can we live without having any intention in mind?
if i consider myself, i think i am not without intention even for a second - even if i sit in meditation
The theory is, there is intention right up to the moment of awakening. In the moment of awakening, there is no intention.
is it really possible to have no intention in mind and live in this world?
No, but the practice does lead to a place where you're not attached to living in this world, so I guess you hold your attentions more lightly at that point. Enlightened people including the Buddha definitely act with intention.
When the founder of a religion explicitly says what the purpose of it is, it is reasonable to construe that as the purpose. When someone else says the purpose is something else and provides no evidence for it, it is reasonable to assume that they don't know what they're talking about.
The point is that all religions are fundamentally the same. They have the same purpose etc. That's probably why they are all categorized as "religion."
You keep saying that, but don't seem to have much of an argument beyond "Yeah, no real difference there," when it's clear that there are significant differences between nirvana and heaven, for instance. I think you have to be more specific about what "fundamentally the same" means, because you seem to have a rather peculiar interpretation of it.
It's not just human birth, it's having reached a point in your life where the dharma makes sense to you, having sufficiently peaceful and bountiful circumstances forr practice, and the guidance to do practice effectively. Much fewer than 1% of people have had those circumstances, and you never know when they're going to end.